In a small, heavy-based saucepan, combine the sugar and 1/4 cup water. Bring to a boil over medium-high heat, and cook until the liquid turns a light caramel color, carefully swirling the pan every few minutes to avoid crystals forming, about 8 minutes. Pour the caramel into a 9- by 2-inch round cake pan or 8- by 4-inch loaf pan, and swirl to coat the bottom and up the sides. Let cool. Place the cake pan into a 3-inch-deep baking pan.

In a medium saucepan, combine the whole milk, evaporated milk, and vanilla. Add the cinnamon, star anise, and lime zest and bring to a boil over medium-high heat. Lower the heat and simmer for 2 to 3 minutes. Strain the mixture into a bowl and discard the solids. Cover with plastic wrap and let cool, 15 minutes.

Heat oven to 350 degrees F. In a large bowl, whisk together the eggs and egg yolks while gradually adding the condensed milk. Add the cooled evaporated milk mixture and stir gently until well incorporated. Place the prepared baking pan on the middle oven rack. Pour the milk mixture into the cake pan. Pour hot water into the baking pan to come halfway up the sides of the cake pan. Bake until just set, 1 hour to 1 hour 10 minutes. Remove from the oven and let cool to room temperature. Chill in the refrigerator, in the pan, for at least 3 hours before serving. Using a knife, loosen the flan around the edges and turn out onto a plate.

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